A Manifold Operator Representation for Adaptive Design ∗

Jacob BealRaytheon BBN TechnologiesCambridge, MA, USA 02138

jakebeal@bbn.com

Hala MostafaRaytheon BBN TechnologiesCambridge, MA, USA 02138

hmostafa@bbn.com

Annan MozeikaiRobot Corporation

Bedford, MA, USA 01730amozeika@irobot.com

Benjamin AxelrodiRobot Corporation

Bedford, MA, USA 01730baxelrod@irobot.com

Aaron AdlerRaytheon BBN TechnologiesCambridge, MA, USA 02138

aadler@bbn.com

Gretchen MarkiewiczRaytheon BBN TechnologiesCambridge, MA, USA 02138

gmarkiew@bbn.com

ABSTRACTMany natural organisms exhibit canalization: small geneticchanges are accommodated by adaptation in other systemsthat interact with them. Engineered systems, however, aretypically quite brittle, making design automation extremelydifficult. We propose to address this problem with a gen-erative representation of design based on manifold opera-tors. The operator set we propose combines the intuitivesimplicity of top-down rewrite rules with the flexibility anddistortion tolerance of bottom-up GRN-based models. Anembryogeny specified using this representation thus placesconstraints on a developing design, rather than specifying afixed body plan, allowing canalization processes to modulatethe design as it continues to develop. We demonstrate ourideas in the domain of electromechanical design and validatethem with simulations at different levels of abstraction.

Categories and Subject DescriptorsD.2.2 [Software Engineering]: Design Tools and Tech-niques; I.2.9 [Artificial Intelligence]: Robotics

General TermsDesign

KeywordsMorphogenetic Engineering, Generative Design, Implicit Em-bryogeny, Generative Representation, Functional Blueprints

1. INTRODUCTIONWhen human engineers make modifications to a design,

not all parameters are treated equally. Instead, a design

∗Sponsored by DARPA under contract W91CRB-11-C-0052; views and conclusions contained in this document arethose of the authors and not DARPA or the U.S. Gov’t.

Permission to make digital or hard copies of all or part of this work forpersonal or classroom use is granted without fee provided that copies arenot made or distributed for profit or commercial advantage and that copiesbear this notice and the full citation on the first page. To copy otherwise, torepublish, to post on servers or to redistribute to lists, requires prior specificpermission and/or a fee.GECCO’12, July 7-11, 2012, Philadelphia, Pennsylvania, USA.Copyright 2012 ACM 978-1-4503-1177-9/12/07 ...$10.00.

(a) miniDroid

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Figure 1: We investigate embryogeny as a represen-tation for adaptable structures using the example ofa “miniDroid” robot (a), beginning by developing abody plan (b) from an undifferentiated egg.

expert typically begins by addressing only the parametersmost critical to the desired modification. Constraints be-tween the elements that are modified and other parts of thedesign then imply other changes, which may ripple through-out the entire design. In fact, human engineers considermany of these implicit changes to not be changes at all, butin fact to be keeping the same design.

This suggests that design evolvability by any automatedsystem, from evolutionary algorithms to knowledge-baseddesign tools, might be greatly enhanced by using a repre-sentation capturing relations exploited by a human designer.This would allow each element to adjust its phenotype forcompatibility with other elements it interacts with.

In natural biological organisms, the processes of morpho-genesis adapt structure to environment remarkably well onboth an individual and evolutionary time scale. Significantlyfor our purposes, small genetic or somatic changes can pro-duce cascading effects during development in other systemsthat interact with them. This ability to accommodate smallchanges without causing related components to fail is termedcanalization [29]. Great strides have been made in under-standing the building blocks of biological adaptivity and theties between morphogenesis and evolution [9, 19], and therehave been a number of systems that apply ideas from mor-phogenesis to particular aspects of design (see Section 2).No clear framework has yet been developed, however, forexploiting morphogenetic principles in the creation of gen-eral engineered systems.

In this paper, we present a generative representation for

electromechanical system designs, using as an example asmall ground robot, the “miniDroid” (Figure 1). Generativerepresentations have the advantages of being compact andcapable of representing highly complex phenotypes. Suchrepresentations are typically implemented either as top-downrewrite rules or bottom-up cellular models. The former hasthe advantage of being more intuitive to craft, while the lat-ter exhibit more tolerance to distortions. Our encoding hasthe best of both worlds because it is rule based, but the rulesinvoke manifold operators that are robust to distortions (ex-ecuting correctly regardless of the geometry).

Our approach promotes canalization; the result of our em-bryogeny is not the final body plan, but a web of constraintsand relations among design parameters. This might be ap-plied to both the form of a body and its controller, thoughin this paper we restrict ourselves to considering only theform. This affords more freedom to the design process toadapt the design by changing key parameters and havingthe rest be derived in accordance with the constraints. Wesuggest that this may allow for more flexible and efficientdesigns, where much of the process of adaptation can becarried out autonomously.

We instantiate our morphogenetic model framework witha candidate basis set of five manifold geometry operators, in-spired by common processes in the embryogeny of animals.Such programs might then be integrated together with func-tional blueprints [1, 3] to facilitate design adaptation. Fi-nally, we use the miniDroid for preliminary validation of thisapproach, creating a program for development of a robotbody plan and testing the approach with simulations at sev-eral levels of abstraction.

2. RELATED WORKAutomated design of electrical and mechanical systems

has long been a topic of interest. While there has beenmuch work in the area (e.g., [8], [16], [14]), it has facedsignificant obstacles from the complexity of searching thespace of possible designs. A primary aim of our work is aprincipled reduction of this design space.

This aim matches the goals of “morphogenetic engineer-ing” as proposed by Doursat [12] and for robotics in [18] and[28]. Initial work towards a framework for morphogeneticengineering has been laid out for evolvable pattern forma-tion in [11]. Meng et al.’s morphogenetic approach [26] tomodular robotics combines a rule-based controller to gen-erate desired patterns with a simulated genetic regulatorynetwork to coordinate the configuration of robot modules.A more formal mathematical model can be found in [25],though the representational consequences are not explored.

An embryogeny is the process of growth that defines howa genotype is mapped to a phenotype [6]. Several worksadvocate the use of indirect mappings (also known in theEvolutionary Computing literature as implicit embryogeny)from genotype to phenotype (e.g., [6], [15], [17]). The geno-type in this case is a generative encoding of the rules thatgovern the development of an individual. Generative en-codings have the advantages of being compact and capableof representing highly complex phenotypes. This is due toparts of the genome being reused during development acrosstime and space. Implicit embryogeny is implemented eitheras top-down rewrite rules or bottom-up cellular models. Insubsequent sections, we elaborate on the similarities betweenour approach and each of these two approaches.

(a) iRobot Warrior (b) iRobot PackBot

(c) iRobot SUGV (d) iRobot LANdroid

Figure 2: Families of engineered systems often ex-hibit “phylogenetic” relationships similar to those ofnatural organisms.

A number of specialized languages for shape formationhave been inspired by morphogenesis. For example, chemi-cal gradients are the basis of Nagpal’s Origami Shape Lan-guage [27], which produces deformable geometric patterns,and Coore’s Growing Point Language [10], which forms topo-logical patterns using a model of plant tropisms. Likewise,Kondacs’ model of pattern formation [21] creates regenera-ble patterns using a model of cell reproduction and apop-tosis. However, such languages have been specialized andhard to apply to the design of realistic engineered systems.

Regardless of the specifics of the representation, geneticalgorithms are typically used as the design process that mod-ifies a genome to achieve a desired form/function. An al-ternative is offered by functional blueprints [1, 3], an engi-neering approach in which a system is specified accordingto its desired performance, with programs to incrementallyadjust structure to improve performance when the goals arenot met. The representation we provide in this paper iswell-suited for interaction with either genetic algorithms orfunctional blueprints.

3. EMBRYOGENY CAN ENCODE DESIGNPARAMETER RELATIONS

In any engineered system, some parameters strongly in-fluence overall system design, while others can be derivedby an expert human designer from the interaction of thekey parameters. To better understand how these sort ofsystematic relations might be exploited, we focus on the ex-ample of electromechanical systems, and in particular thethe family of products that the iRobot corporation has de-rived from their original PackBot system (Figure 2), and anew“miniDroid”design based on the same architecture (Fig-ure 1(a)). Although these robots span a wide range of sizesand applications, all of them share a base body plan, includ-ing paired treads each driven by two wheels, flippers coax-ial with one wheel, and a top-mounted sensor/manipulatorpackage. The loose similarity to natural phylogenetic fami-lies suggests that the natural world may yield hints towarda design representation that facilitates such variation.

Consider a simple modification to the miniDroid which,like all robots in the PackBot family, uses its flippers to climbover obstacles. When a designer increases the miniDroid’s

length

length

(a) Part-Centric

length

length

(b) Interface-Centric

Figure 3: Coordinate systems can implicitly encodedesign: interface-centric coordinates let a miniDroidflipper extend outward when lengthened.

flipper length (e.g., to climb over slightly taller obstacles)should the flipper extend symmetrically around its geomet-ric center (Figure 3(a)), away from its attachment point(Figure 3(b)), or some combination thereof? The answeris intuitive to a human designer, but a self-adapting designmust have some way of representing this relationship.

This is potentially a severe problem: engineered designshave extremely high numbers of parameters: even a simplebeam is described by at least nine parameters (side lengths,position, and orientation). Only a few of these parametersare key, while most are constrained by their relationship tokey parameters. Natural systems, however, must solve thisproblem in the embryogenies that create their forms, sinceevolution acts through relatively independent incrementalchanges to very few parameters at a time.

In animals, the process of embryogenesis effectively spec-ifies a hierarchy of spatial relationships and coordinate sys-tems that constrain how changes can occur. For example,when an arm is lengthened, it extends further out from thebody, rather than attempting to invade into the body.

We thus look to embryogeny as a means of encoding re-lationships between design parameters, defining it as a par-tially ordered sequence of operations over a spatial computerthat convert a simple initial “egg” into a “mature” structure.It is important to note that although the systems we designare not going to be physically built using a developmentalprocess, we still need to simulate this process because itgenerates by-products (e.g., various coordinate systems lo-cal to tissues, tissue specialization hierarchies, and simulatedphysical interactions between tissues) that guide the main-tenance of a design’s integration as its components are beingchanged, whether by a human designer, an evolutionary al-gorithm, or an online process for repair and adaptation.

4. DEVELOPMENTAL REPRESENTATIONSWe now propose a representation of embryogeny using a

candidate basis set of manifold geometry operators, thendemonstrate how this can be applied to develop the bodyplan of a miniDroid. For reasons of space, we give only ahigh-level description of the operators and the implementa-tions we have constructed. Full details can be obtained fromthe MADV release online at http://madv.bbn.com.

4.1 Manifold OperatorsIn natural systems, a vast array of mechanisms are used

to develop the form of an organism. From this panoply,we extract five simple abstract manifold geometry operatorsfor measuring, segmenting, and modifying manifolds. Thisis not necessarily the “best” set of operators, but for pre-liminary work, we minimize the complexity of the model inorder to simplify analysis of its capabilities and implications.

We formulate these as operators on continuous manifolds,rather than simpler Euclidean geometric operators, so thatthey can tolerate topological distortions. Thus when a struc-ture’s shape and size are changed (either by changes toits antecedents or by physical interaction with other struc-tures), the operators should still produce qualitatively sim-ilar effects, promoting canalization. These operators can beapplied to either the entire structure or to any substructure(viewed as a submanifold), and affect surrounding tissuesonly indirectly, as a byproduct of physical interaction.

Figure 4 illustrates our set of manifold operators:

• coordinatize(0-region, 1-region): This operator com-putes a local coordinate system between two regions,ranging from 0 in one region linearly to 1 in the otherregion. It is based on the use of chemical gradients topolarize tissues, and most particularly on those wherethe signal is the ratio of two chemicals.

• latch(region, type): The latch operator differenti-ates a region into a component type. This operator isanalogous to determination of cell fate in a biologicalsystem, such as the determination of limb fields.

• scale(region, scale-factor): The scale operator takesa region and increases or decreases its scale proportion-ately along one or more axes. This is inspired by di-rected proliferation, such as is used in the constructionof limb buds.

• connect(source-region, destination-region): Thisoperator extends a new region from a source regionalong a near-shortest path to a destination region. Itis based on cell migration, particularly the migrationof tissue sheets, and on the extension of cell processessuch as axons. We have tentatively specified that onlythe closest point in the source will connect, and thatthe width of the connection will be proportional to thesize of the smaller region.

• speckle(region, expected-separation): The speckleoperator selects a set of small regions filling the space,each region determined randomly and separated by anexpected distance from all others nearby. This is in-spired by symmetry-breaking processes, such as thoseused to create hair follicles.

When composed using arithmetic, branching, and func-tion definition, we believe that this set of manifold opera-tors may be sufficient to express most arbitrarily complexadaptable structures (though they are not space-time uni-versal [2]). Many other biological mechanisms can also beexpressed as composites of these operators: for example, re-cruitment can be expressed as coordinatize followed by latch.

4.2 Example: miniDroid Body PlanTo validate the plausibility of our candidate operators, we

have created a draft developmental sequence for the bodyplan of a miniDroid, which is likely to also apply well to othermembers of the PackBot family. The sequence begins witha cuboid egg (to better match current engineered systemsand manufacturing processes), and uses all operators exceptfor speckle. Our developmental sequence can be viewed interms of eight stages, shown in Figure 5, though executionmay proceed somewhat asynchronously:

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Figure 4: Candidate manifold geometry operators to be a basis set for developing electromechanical designs.

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Figure 5: Approximate developmental stages of a miniDroid (or other PackBot family member) body plan.

• Stage 1: basal coordinates are created, along antero-posterior, dorsoventral, and mediolateral axes. Oper-ators: coordinatize

• Stage 2: basal coordinates are used to partition therobot into coarse body plan, with the exterior latchinginto “skin.” Operators: latch

• Stage 3: the electronics region differentiates along theanteroposterior axis into sensor and CPU regions. Theboundary between limb buds recruits nearby materialto form a gap between anterior and posterior limbs.Operators: coordinatize, latch

• Stage 4: limb buds scale themselves out of the body,then establish a proximodistal axis and local centro-medial axis. The power region recruits inter-limb gapmaterial. Operators: coordinatize, latch, scale

• Stage 5: distal section of limb buds differentiates intowheels. Proximal section of front limb buds differenti-ates into drive. For rear limb buds, the flipper axle dif-ferentiates using the centromedial axis, and the prox-imal section of the limb bud differentiates into a pas-sive mounting point. The wheels will round themselveslater, as the body scales up. Operators: latch

• Stage 6: flipper axles scale in and out, meeting in thecenter. Wheel edges connect tracks between wheels.Operators: scale, connect, latch

• Stage 7: flipper axles scale outwards to form the flip-per drive, pressing away nearby sections of CPU andpower. Distal sections of axle differentiate into flipperbuds and form a new anteroposterior axis. Opera-tors: scale, coordinatize, latch

• Stage 8: flippers scale from buds. Operators: scale

At Stage 8, the complete body plan has been formed.Each of the sections can now refine its details, while scalingup to reach the mature size and component scale relationsspecified by its genotype. The connectivity relations devel-oped in the body plan constrain this scaling process, resolv-ing implicit parameters: for example, if the flipper grows

longer it can only do so by lengthening the portion not at-tached to the axle, and if the overall body grows wider, thewheel and motor assemblies will be carried outward alongwith the expansion. Meanwhile, power and signal wires con-nect the CPU, batteries, and motors using the connect op-erator, a process similar to innervation in vertebrates.

Together, these processes should produce a design similarto that of the miniDroid. Integration with a (semi)auto-nomous design process, such as an evolutionary algorithmsor functional blueprints, should be possible by placing thescaling parameters and/or embryogeny under control of thedesign process.

5. IMPLEMENTATIONTo begin validation of our approach, we have implemented

simulations at four different levels of abstraction:

• At the highest level of abstraction, a tissue-level simu-lator demonstrates flexible application of manifold op-erators using an asynchronous rule system.

• Refining to a cellular model, we demonstrate that us-ing manifold operators allows a design to maintain co-herent inter-component relations even if the adapta-tion process subjects it to severe distortions.

• A soft-body simulation demonstrates the ability to com-pute adhesion, displacement, and penetration of com-ponents at reasonable computational cost, enablingphysics-based resolution of component layout.

• Finally, adaptive refinement of details under feedbackcontrol is demonstrated with a chemotactic wiring model.

Each of these demonstrates a different aspect of how ourmorphogenetic approach enables canalization of a develop-ing design. Having demonstrated that each requirement ofthe system is individually reasonable, the next obvious stepwill be to integrate them to produce a single system thatcompletely constrains the realization of design phenotypes.

5.1 Tissue-Level ModelsIn the previous section, we proposed a set of manifold op-

erators for representing development. The tissue-level model

(a) Initial robot body (b) Partially developed

(c) Four wheeled robot (d) Eight wheeled variant

Figure 6: (a) The initial robot body is (b) updatedusing the set of development rules to create either(c) a four or (d) eight wheeled robot (wheels areunrounded green boxes).

demonstrates how these can be combined into a flexible em-bryogeny through asynchronous rule execution.

For this simulation, we implemented a prototypical spec-ification of embryogeny, executor, and visualizer in Java.The simulation uses a coarse-grained representation of cellcollectives (tissues), where all cells carry out the same func-tion, analogous to biological tissues. Each tissue contains aset of body part labels (e.g., “flipperBud”), a geometry spec-ifying its shape, and a means of converting between its owncoordinate system and others.

The tissues are manipulated by a set of developmentalrules. Our choice of asynchronous rules for a representationis inspired by naturally occurring genetic regulatory net-works (GRNs): both of these have the useful property of sep-arating execution conditions (promoter sequence of GRNs,preconditions of rules) from actions (encoded protein/nucleic-acid complexes of GRNs, post-conditions of rules). The rulesare thus loosely coupled, reducing the need for explicit or-dering constraints on their execution. As in other generativedesign systems, iteration and conditional execution are im-plicit and arise out of the interaction of the rules.

In our representation, a rule applies to any tissue thatmatches the rule’s preconditions. An executor starts firingthe developmental rules on an undifferentiated robot body(‘egg’), breaking ties in arbitrary order. As rules differen-tiate and manipulate parts of the body, other rules becomeapplicable. A rule’s effects are a sequence of manifold oper-ators from the basis set given in 4.1.

A difference between our developmental rules and tradi-tional re-write rules is that ours operate in the context ofa physics simulator that places physical constraints on theireffects. Another important difference is that our rules canhave non-local triggers and effects (e.g., a rule involving scal-ing can result in attached tissues being dragged along).

We have implemented a miniDroid embryogeny, whichuses 12 rule skeletons that are parameterized into 23 rulesthat carry out the 8 developmental stages described in theprevious section. Currently, our rules avoid operations thatwould require interactions between tissues, e.g., penetrating

other tissue. Our work to support such interactions is de-scribed in Sections 5.2 and 5.3. The rules produce a bodyplan of a robotic “embryo” made up of various “organs” likelimbs, flippers and wheels. We were careful to keep the rulesoblivious to absolute dimensions and positions, a task thatwas facilitated by the use of the manifold operators. Fig-ure 6 shows the initial body (6(a)) and the body-plan afterembryogeny (6(c)). An example of the adaptability of ourencoding is shown in Figure 6(d), which shows an eight-wheeled variant created by changing the single parameterspecifying the number of body segments (pairs of wheels).

The body plan is not yet the final design, but effectivelyencodes constraints on the design parameters, allowing manyimplicit parameters to be resolved by means of the geometricand topological relationships established in the elaboratedmanifold developed by executing these rules. For example,the CPU and batteries are inside the body, though theirrelative positions within are not fixed, and the flippers areattached at the axle, so if they are made longer, they canonly grow outward away from the axle.

Although rule-based generative encodings have been widelyused (e.g., [17], [20], [23]), we note that our encoding and theresulting phenotypes are very different from those of priorsystems. Existing approaches develop systems consisting oflarge numbers of simple repeated structures. For example,in Framsticks [20], the body of a robot is composed of a setof interconnected simple elements called sticks. The simplic-ity, uniformity and large numbers of these system buildingblocks allow the rewrite rules significant freedom in manip-ulating the structure. Most electromechanical designs, how-ever, like the miniDroid, are made of parts that are highlyheterogeneous and tightly coupled. Our rules therefore can-not arbitrarily rewrite the body, but need a representationlike ours, which allows parts to maintain particular physicaland topological relationships as they differentiate.

5.2 Cellular ModelsComputing arbitrary distortion of manifolds at the tissue

level is extremely challenging due to the difficulty of analyt-ically representing and manipulating arbitrary geometries,so we demonstrate the distortion tolerance of manifold op-erators using a cellular simulation instead. We implementedthis model in the Proto [4] spatial computing language. InProto, a program is specified in terms of geometric compu-tations and information flow on a continuous manifold, thencompiled to generate a local program that approximates theglobal specification. Because our proposed manifold opera-tors were deliberately selected to be compatible with suchtransformations, the local programs produced from them aresimple and could be readily transformed into a GRN rep-resentation.1 Proto thus links tissue-level representations,such as those discussed in the previous section, with cellularmodels where it is simpler to represent approximate topo-logical distortions.

We have implemented the first five stages of the miniDroidembryogeny, from the initial coordinatization of the initialbody through the scaling and differentiation of limb buds.While the Proto implementation does not represent rules(we instead transform the dependencies into flow control),its manifold representation allows the program to executeunder distortion on a wide range of alternate manifolds—a

1In fact, Proto programs have already been transformed intoactual biological GRNs [5].

(a) Base body (b) Stage 1/2/3 overlap

(c) Stage 3 (d) Stage 4

(e) Stage 5 (f) Stage 5, side view

Figure 7: Stages 1 to 5 of body-plan developmentexecuted in fine-grained Proto models. From theinitial body (a), the spreading coordinate systemstrigger differentiation (b) into the base components(c). The limb buds then scale outward (d) and dif-ferentiate into wheels, motors, and mounts (e,f).

critical capability for emulating the adaptivity exhibited bynatural morphogenesis. The continuous model of distributedexecution also provides a natural path toward using blendingto resolve conflicts between rules.

Figure 7 shows snapshots of embryogeny executing on amodel of 2000 cells distributed through a rectilinear 3D vol-ume of space. Figure 8 compares the result of execution thesame program with different initial conditions, showing theinherent geometric adaptation of the program.

Notice that the operators of the embryogeny are able toexecute both asynchronously and simultaneously, and thatthe manifold representation allows the program to producesensible results even on a radically distorted underlying man-ifold. The mathematical nature of manifold geometry thusprovides one form of canalization: whenever a region ofspace is transformed, the coordinate fields created by coor-dinatize operators transform along with it, and any subse-quent operator that is applied to that region is transformedto match as a consequence.

Although our model is “cellular,” it is very different fromthe cellular models discussed in works on implicit embryo-geny, which tend to be more bio-mimetic, capturing low-level biological processes like GRNs and cell-level interac-

Figure 8: Proto’s manifold model allows embryo-geny to automatically adapt to execute on changedunderlying spaces. Shown (l to r) are execution ona flat egg, an ovoid egg, and twisted coordinates.

tions (e.g., the work by Eggenberger [13] implements con-cepts like cell division, death and induction at the cell andGRN level). The question often addressed in the literatureis how to go from low-level interactions to the desired globalbehavior/structure. The answer typically relies on an exter-nal element (e.g., fitness function in a genetic algorithm) toassess the collective behavior/structure arising out of the in-teractions. This assessment is then used as a guide to makechanges to the low-level interactions that hopefully (thoughnot directly) move the system closer to its goal. Our ap-proach differs in that in our cellular model, we compile thedesired global behaviors into low-level interactions, insteadof the other way around. This greatly simplifies the prob-lem, allowing us to develop more complex morphologies thanthose in the literature (e.g., [7], [13], [22]).

5.3 Soft-Body Tissue InteractionIn many cases, developing components may try to move

relative to connected components or to move into occupiedspace. The adaptability of our manifold representation al-lows many such conflicts in design layout to simply be re-solved physically, by having tissues push, pull, shear, andpenetrate one another, much like biological tissues. Thispromotes canalization by allowing structures to develop ina dynamically adjusted physical context (using the distor-tion tolerance discussed in the previous section), rather thanforcing all geometric conflicts to be resolved in the develop-mental program. For example, in Stage 6 of miniDroid de-velopment, the flipper axles scale outward, penetrating thewheels (Figure 5(f)), which means that the wheel programneed not be modified to allow for the development of flip-pers.

Implementing this interaction requires a soft-body simu-lation, which must be computationally inexpensive enoughto be practical. We are not attempting to simulate tissueinteractions in a way that is necessarily biologically plausi-ble. We need only to represent tissues in a way that allowsthem to flexibly interact during growth operations.

As a first case study we simulated a tissue scaling operatorpenetrating into an adjacent tissue with the focus being onhow the penetrated tissue responds (Figure 9), analogous tothe flipper axle/wheel interaction described above. The axlemust push up against the inner walls of the wheels; the wheelgives way elastically up to a point and then begins to plas-tically deform, allowing the flipper axle to penetrate. Afterinvestigating several representations of tissues, node basedtissue representations have produced the most desirable re-sults. Nodes (cells) were uniformly distributed throughoutthe volume contained by a surface mesh (initial state). Thecells are outfitted with several interactions which allow thetissue to retain shape when under no external forces (from

(a) Initial State (b) Deformation (c) Penetration

Figure 9: Soft-body model of a shaft growing axiallybetween two wheels (a), that deform outward (b),then are penetrated and deform plastically (c).

adjacent tissues) while allowing deformation to occur (plas-tically and elastically) when external forces are applied. Alow resolution representation of the miniDroid has been im-plemented using the node based tissue representation.

Simulations were implemented using MASON, a Java-based multi-agent simulator [24]. Our interaction models usedirected growth, implemented by assigning a growth vectorλ to each class of cells (tissue type). The growth rate foreach cell is computed from a neighborhood vector η, whichwe define as the distance-weighted centroid of the N cellswithin distance M of the position of the current cell:2

η =

N∑i

x− xi

||x− xi||(1)

where x is the position of the current cell, xi is its ith neigh-bor, and || || denotes vector length. The growth, or aging,

rate for a particular cell, Γ, is the dot product of η and λ̂:

Γ = η · λ̂ (2)

where λ̂ is the normalized vector. A cell splits to create achild when its age reaches a certain threshold. The child isplaced at a random location in a sphere of radius λ, centeredaround the parent. Then the parent’s location adjusted sothat their centroid of the pair matches the parent’s originallocation.

Directed growth is not enough to maintain tissue shape,so intercellular interactions are also directed according to λ.We modeled three inter-cellular forces: 1) a strong repellentforce when two cells are overlapping, 2) a weak attractiveforce between cells of the same class within some small dis-tance, and 3) a weak repellent force between cells of differentclasses within some small distance. All three of these forcesare governed with the same underlying equation:

ftypeij = (xi − xj)−fmaxtype

(ri + rj)+ fmaxtype (3)

where ri and rj are the two cell radii, and fmaxtype is themaximum value for that force type.

We found these forces to be necessary to keep groups ofthe same cell type together, and prevent too much mixingbetween types. These forces are then summed over neigh-bors as follows:

F =

N∑i

((f1i + f2i + f3i) · λ̂)(x− xi)

||x− xi||(4)

2We found stability was improved by limiting the maximumnumber of neighbors in the calculation.

Figure 10: Wiring model executing in a simulatedminiDroid interior. Components are modeled asrectangles, with wires routing around obstacles.

This ensures that motion due to inter-cellular forces is morepronounced in the directions that align with λ.

Finally, we also ensure that cells that do not become toodense by killing off cells with high forces exerted on them.This also aids in deformation and penetration when cellclasses collide.

Our simulation method is efficient, requiring only O(nN)computation, where n is the number of cells and N theneighborhood size, but can still be much improved.

5.4 Wiring ModelsThe connect operator enables another form of canaliza-

tion that is also quite important in biological organisms:the integration of systems that initially develop separately.An important example of this is the chemotactic process bywhich neurons grow axons to reach appropriate target cellsin an organism’s developing nervous system. Because this in-tegration happens as a feedback process, perturbations suchas congenital extra fingers or toes are simply integrated intothe system and become functional.

With the miniDroid, wiring components to the CPU andpower supply presents an analogous problem. We imple-mented a self-stabilizing connect operator in Proto, and usedit to implement autonomous wiring design for the miniDroid.The connect operator is implemented by having its destina-tion emit a chemical whose diffusion forms a gradient thatpropagates through penetrable tissue only (i.e., around solidobstacles like motors). The source follows this gradient backto the destination, forming a near-shortest path betweenthem. Because the computation is self-stabilizing, when thestructure changes the path automatically adjusts as well.

The wiring model for the miniDroid uses 9 connect opera-tors to create 15 wires connecting the motors, CPU, batter-ies, and flipper encoder. Figure 10 shows the wiring modelexecuting on a 4,000 cell approximation of a miniDroid inte-rior, in which wires successfully connect signal, power, andground between the batteries, CPU, encoder, and motors.

This demonstrates a fourth form of canalization enabledby our manifold operators, in which a functional blueprintacts during embryogeny to ensure functionality of a designby actively modifying it toward specified goals as it develops.

6. CONTRIBUTIONS AND FUTURE WORKWe have presented a morphogenetic engineering frame-

work for simplifying design adaptation. This simplifica-tion is due to implicit design parameters being derived fromkey parameters, made possible by encoding the design con-straints in our implicit embryogeny. We also proposed a

basis set of manifold operators for representing distortion-tolerant embryogeny, and applied them to the example ofthe miniDroid. We further demonstrated the plausibility ofthis approach with four simulations at various levels of ab-straction: models of tissue-level development, cellular devel-opment under distortion, soft-body component interaction,and wiring design. One measure of success of our designtool is comparing the size of the design space (e.g., in termsof the number of parameters a design process can manip-ulate) to the size of the space explored by our framework.Preliminary results show a dramatic decrease in the searchspace.

Three clear directions for future work follow from the workpresented in this paper: first, integration of our simulationsto produce a complete mapping from genotype to adaptedphenotype. Second, integration with (semi)autonomous de-sign adaptation methods such as evolutionary algorithms orfunctional blueprints, and quantification of the benefits ofincorporating morphogenetic constraints. Finally, and on amore general level, theoretical investigation of how use ofmorphogenetic models modifies a fitness landscape.

7. ADDITIONAL AUTHORSAdditional authors: Kyle Usbeck (Raytheon BBN Tech-

nologies, email: kusbeck@bbn.com)

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